Theorem xpdom2g 7123

Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
xpdom2g	|- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Proof of Theorem xpdom2g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4786	. . . . 5 |- ( x = C -> ( x X. A ) = ( C X. A ) )
2		xpeq1 4786	. . . . 5 |- ( x = C -> ( x X. B ) = ( C X. B ) )
3	1, 2	breq12d 4141	. . . 4 |- ( x = C -> ( ( x X. A ) ~<_ ( x X. B ) <-> ( C X. A ) ~<_ ( C X. B ) ) )
4	3	imbi2d 230	. . 3 |- ( x = C -> ( ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) ) <-> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) ) )
5		vex 2824	. . . 4 |- x e. _V
6	5	xpdom2 7122	. . 3 |- ( A ~<_ B -> ( x X. A ) ~<_ ( x X. B ) )
7	4, 6	vtoclg 2883	. 2 |- ( C e. V -> ( A ~<_ B -> ( C X. A ) ~<_ ( C X. B ) ) )
8	7	imp 124	1 |- ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4128   X. cxp 4770   ~<_ cdom 7014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fv 5383  df-dom 7017

This theorem is referenced by:  xpdom1g  7124